In semiconductor processing, it has become more and more important and also difficult to characterize material properties along with the continuous down-scaling of the critical dimensions. Sheet materials such as ultra-shallow junctions are widely used, and sheet resistance measurement has been developed for the study of these. In addition to the sheet resistance of film materials, the sheet carrier density and the mobility are also important properties for the performance of semiconductor device, e.g. in complementary metal-oxide-semiconductor (CMOS) transistors. However, most of the present measurement methods for determining the sheet resistance, the sheet carrier density, and the sheet carrier mobility require either special sample preparation or destructive machining of the sample. Some measurement methods also require several engagements between a probe and a test sample, which impairs the efficiency of the methods.
In order to characterize the sheet carrier density and the mobility of sheet materials, a fast, cheap, and non-destructive method is therefore desired. There are several methods employing microscopic multi-point probes produced by silicon-based micro-fabrication technology that can be used for the sheet resistance measurements. In WO2007045246A1 it has been shown that Hall effect measurement can be made with an in-line micro four-point probe to measure the sheet resistance, the sheet carrier density, and the Hall mobility. However, the method demands two measurement points, i.e. two different engagements between probe and test sample, which can cause large measurement errors when the electrical property is not ideally homogeneous across the sample under test.
It is an object of the present invention to improve the accuracy and reduce the time of determining an electrical property of a test sample. In particular, it is an object of the present invention to improve the accuracy and reduce the time of determining an electrical property that depends on the distance to an electrical boundary of the test sample. According to the present invention, only one measurement point, i.e. one engagement between probe and test sample, is necessary in a Hall effect measurement. This will reduce the measuring time significantly and provide more accurate results for electrical transport properties of thin films.
The object of the invention is achieved according to a first aspect of the invention by a first position on an electrically conducting surface portion of a test sample and an electrical boundary of the electrically conducting surface portion by a multi-point probe comprising a first contact element, a second contact element, a third contact element, and a fourth contact element, each contact element defining a contact point for establishing an electrical contact with the test sample, the method comprising: (i.i) contacting the test sample with the first contact element, the second contact element, the third contact element, and the fourth contact element at the first position on the electrically conducting surface portion, (i.ii) applying a magnetic field having a major field component perpendicular to the electrically conducting surface portion at the first position, (i.iii) applying a first electrical potential across the first and third contact elements for generating a first current in the surface portion at the first position, (i.iv) measuring the first current through the first or the third contact element, (i.v) measuring a first voltage across the second and fourth contact elements, (i.vi) calculating a first resistance value (RB) based on the first current and the first voltage, (i.vii) applying a second electrical potential across the second and fourth contact elements for generating a second current in the surface portion at the first position, (i.viii) measuring the second current through the second or the fourth contact element, (i.ix) measuring a second voltage across the first and third contact elements, (i.x) calculating a second resistance value (RB′) based on the second current and the second voltage, (i.xi) calculating a first resistance difference (ΔRBB′) based on the difference between the first resistance value and the second resistance value, (i.xii) applying a third electrical potential across the first and second contact elements for generating a third current in the surface portion at the first position, (i.xiii) measuring the third current through the first or the second contact element, (i.xiv) measuring a third voltage across the third and fourth contact elements, (i.xv) calculating a third resistance value (RC) based on the third current and the third voltage, (i.xvi) applying a fourth electrical potential across the third and fourth contact elements for generating a fourth current in the surface portion at the first position, (i.xvii) measuring the fourth current through the third or the fourth contact element, (i.xviii) measuring a fourth voltage across the first and second contact elements, (i.xix) calculating a fourth resistance value (RC′) based on the fourth current and the fourth voltage, and (i.xx) calculating a second resistance difference (ΔRCC′) based on the difference between the third resistance value and the fourth resistance value, or in an alternative replacing steps (i.xii) to (i.xx) with: the multi-point probe comprising a plurality of contact elements, each contact element defining a contact point for establishing an electrical contact with the test sample, the plurality of contact elements comprising the first contact element, the second contact element, the third contact element, the fourth contact element and one or more additional contact elements, (ii.xii) defining a first configuration of contact elements of the plurality of contact elements, the first configuration of contact elements being composed of the first contact element, the second contact element, the third contact element, and the fourth contact element, (ii.xiii) defining a second configuration of contact elements of the plurality of contact elements, the second configuration of contact elements being composed of a fifth contact element, a sixth contact element, a seventh contact element, and an eighth contact element, at least one contact element of the second configuration of contact elements being a contact element of the one or more additional contact elements, (ii.xiv) contacting the test sample with the contact elements of the second configuration of contact elements at the first position on the electrically conducting surface portion at the same time as contacting the test sample with the contact elements of the first configuration of contact elements, (ii.xv) applying a third electrical potential across the fifth and seventh contact elements for generating a third current in the surface portion at the first position, (ii.xvi) measuring the third current through the fifth or the seventh contact element, (ii.xvii) measuring a third voltage across the sixth and eighth contact elements, (ii.xviii) calculating a third resistance value (RB,2) based on the third current and the third voltage, (ii.xix) applying a fourth electrical potential across the sixth and eighth contact elements for generating a fourth current in the surface portion at the first position, (ii.xx) measuring the fourth current through the sixth or the eighth contact element, (ii.xxi) measuring a fourth voltage across the fifth and seventh contact elements, (ii.xxii) calculating a fourth resistance value (RB′,2) based on the fourth current and the fourth voltage, and (ii.xxiii) calculating a second resistance difference (ΔRBB′,2) based on the difference between the third resistance value and the fourth resistance value, and both the alternative including the steps (i.xii) to (i.xx) and the alternative including the steps (ii.xii) to (ii.xxiii) further comprising: (i.xxi) defining a first relation (f) including a first, a second, and a third parameter representing the first resistance difference, the second resistance difference, and the distance between the first position and the electrical boundary, respectively, and (i.xxii) employing the first and second resistance differences (ΔRBB′, ΔRCC′, ΔRBB′,2) as the first and second parameters, respectively, in the first relation for determining the third parameter representing the distance (y) between the first position and the electrical boundary.
The method according to the first aspect of the present invention and the alternative replacing steps (i.xii) to (i.xx) may further comprise: (iii.xxiv) calculating a first resistance mean (RBB′,1) of the first resistance value (RB) and the second resistance value (RB′), (iii.xxv) calculating a second resistance mean (RBB′,2) of the third resistance value (RB,2) and the fourth resistance value (RB′,2), and (iii.xxvi) applying a fifth electrical potential across the first and fourth contact elements for generating a fifth current in the surface portion at the first position, (iii.xxvii) measuring the fifth current through the first or the fourth contact element, (iii.xxviii) measuring a fifth voltage across the second and third contact elements, (iii.xxix) calculating a fifth resistance value (RA,1) based on the fifth current and the fifth voltage, (iii.xxx) applying a sixth electrical potential across the second and third contact elements for generating a sixth current in the surface portion at the first position, (iii.xxxi) measuring the sixth current through the second or the third contact element, (iii.xxxii) measuring a sixth voltage across the first and fourth contact elements, (iii.xxxiii) calculating a sixth resistance value (RA′,1) based on the sixth current and the sixth voltage, (iii.xxxiv) calculating a third resistance mean (RAA′,1) of the fifth resistance value (RA,1) and the sixth resistance value (RA′,1), and (iii.xxxv) applying a seventh electrical potential across the fifth and eighth contact elements for generating a seventh current in the surface portion at the first position, (iii.xxxvi) measuring the seventh current through the fifth or the eighth contact element, (iii.xxxvii) measuring a seventh voltage across the sixth and seventh contact elements, (iii.xxxviii) calculating a seventh resistance value (RA,2) based on the seventh current and the seventh voltage, (iii.xxxix) applying an eighth electrical potential across the sixth and seventh contact elements for generating an eighth current in the surface portion at the first position, (iii.xl) measuring the eighth current through the sixth or the seventh contact element, (iii.xli) measuring an eighth voltage across the fifth and eighth contact elements, (iii.xlii) calculating an eighth resistance value (RA′,2) based on the eighth current and the eighth voltage, (iii.xliii) calculating a fourth resistance mean (RAA′,2) of the seventh resistance value (RA,2) and the eighth resistance value (RA′,2), (iii.xliv) defining a second relation including a fourth, fifth, and sixth parameter representing the first resistance mean (RBB′,1), the third resistance mean (RAA′,1), and a first pseudo sheet resistance (RP,1), respectively, (iii.xlv) employing the first resistance mean (RBB′,1) and the third resistance mean (RAA′,1) as the fourth parameter and the fifth parameter, respectively, in the second relation for determining the sixth parameter representing the first pseudo sheet resistance (RP,1), (iii.xlvi) defining a third relation including a seventh, eighth, and ninth parameter representing the second resistance mean (RBB′,2), the fourth resistance mean (RAA′,2), and a second pseudo sheet resistance (RP,2), respectively, (iii.xlvii) employing the second resistance mean (RBB′,2) and the fourth resistance mean (RAA′,2) as the seventh parameter and the eighth parameter, respectively, in the third relation for determining the ninth parameter representing the second pseudo sheet resistance (RP,2), (iii.xlviii) defining a fourth relation (gD) including a tenth, an eleventh, and a twelfth parameter representing the first pseudo sheet resistance (RP,1), the second pseudo sheet resistance (RP,2), and the distance between the first position and the electrical boundary, respectively, and (iii.xlix) employing the first and the second pseudo sheet resistances (RP,1, RP,2) as the tenth and eleventh parameter, respectively, in the fourth relation (gD) for determining the twelfth parameter representing an additional distance (y) between the first position and the electrical boundary, or in an alternative replacing step (iii.xlix) with: the contact elements of the second configuration representing an additional distance (y2) between the first position on the electrically conducting surface portion of the test sample and the electrical boundary of the electrically conducting surface portion, the first relation (fD) further including a thirteenth parameter representing the additional distance between the first position and the electrical boundary, the fourth relation (gD) further including a fourteenth parameter representing the additional distance between the first position and the electrical boundary, and the method further comprising: (iv.xlix) employing the first and the second pseudo sheet resistances (RP,1, RP,2) as the tenth and eleventh parameter, respectively, in the fourth relation (gD) for simultaneously determining the thirteenth parameter and the fourteenth parameter representing the distance (y) and the additional distance (y2) between the first position and the electrical boundary, respectively.
In the alternative including step (iii.xlix) the fourth relation may be equivalent to RP,1/RP,2=gD(y), where RP,1 represents the first pseudo sheet resistance, RP,2 represents the second pseudo sheet resistance, and gD represents a function including the distance y as a parameter, the function gD defining a peak value at a specific distance, and the function gD increasing as a function of the distance below the specific distance and decreasing as a function of the distance above the specific distance, the method may further comprise: (iii.xlx) comparing the distance and the specific distance to determine if the additional distance (y2) is below or above the specific distance in the fourth relation. The additional distance (y2) determined this way is typically more accurate than the distance (y), since it is derived from resistance means instead of resistance differences.
The additional distance may replace the distance for any purpose, e.g. for determining an electrical property of a test sample.
The objects are according to a second aspect of the present invention obtained by a method for determining a distance between a first position on an electrically conducting surface portion of a test sample and an electrical boundary of the electrically conducting surface portion by a multi-point probe comprising a plurality of contact elements, each contact element defining a contact point for establishing an electrical contact with the test sample, the plurality of contact elements comprising a first contact element, a second contact element, a third contact element, a fourth contact element and one or more additional contact elements, the method comprising: (v.i) defining a first configuration of contact elements of the plurality of contact elements, the first configuration of contact elements being composed of the first contact element, the second contact element, the third contact element, and the fourth contact element, (v.ii) defining a second configuration of contact elements of the plurality of contact elements, the second configuration of contact elements being composed of a fifth contact element, a sixth contact element, a seventh contact element, and an eighth contact element, at least one contact element of the second configuration of contact elements being a contact element of the one or more additional contact elements, (v.iii-iv) contacting the test sample with the contact elements of the first and second configurations of contact elements at the first position on the electrically conducting surface portion, (v.v) applying a magnetic field having a major field component perpendicular to the electrically conducting surface portion at the first position (v.vi) applying a first electrical potential across the first and third contact elements for generating a first current in the surface portion at the first position, (v.vii) measuring the first current through the first or the third contact element, (v.viii) measuring a first voltage across the second and fourth contact elements, (v.ix) calculating a first resistance value (RB) based on the first current and the first voltage, (v.x) applying a second electrical potential across the second and fourth contact elements for generating a second current in the surface portion at the first position, (v.xi) measuring the second current through the second or the fourth contact element, (v.xii) measuring a second voltage across the first and third contact elements, (v.xiii) calculating a second resistance value (RB′) based on the second current and the second voltage, (v.xiv) calculating a first resistance mean (RBB′,1) of the first resistance value (RB) and the second resistance value (RB′), (v.xv) applying a third electrical potential across the fifth and seventh contact elements for generating a third current in the surface portion at the first position, (v.xvi) measuring the third current through the fifth or the seventh contact element, (v.xvii) measuring a third voltage across the sixth and eighth contact elements, (v.xviii) calculating a third resistance value (RB,2) based on the third current and the third voltage, (v.xix) applying a fourth electrical potential across the sixth and eighth contact elements for generating a fourth current in the surface portion at the first position, (v.xx) measuring the fourth current through the sixth or the eighth contact element, (v.xxi) measuring a fourth voltage across the fifth and seventh contact elements, (v.xxii) calculating a fourth resistance value (RB′,2) based on the fourth current and the fourth voltage, (v.xxv) calculating a second resistance mean (RBB′,2) of the third resistance value (RB,2) and the fourth resistance value (RB′,2), and (v.xxvi) applying a fifth electrical potential across the first and fourth contact elements for generating a fifth current in the surface portion at the first position, (v.xxvii) measuring the fifth current through the first or the fourth contact element, (v.xxviii) measuring a fifth voltage across the second and third contact elements, (v.xxix) calculating a fifth resistance value (RA,1) based on the fifth current and the fifth voltage, (v.xxx) applying a sixth electrical potential across the second and third contact elements for generating a sixth current in the surface portion at the first position, (v.xxxi) measuring the sixth current through the second or the third contact element, (v.xxxii) measuring a sixth voltage across the first and fourth contact elements, (v.xxxiii) calculating a sixth resistance value (RA′,1) based on the sixth current and the sixth voltage, (v.xxxiv) calculating a third resistance mean (RAA′,1) of the fifth resistance value (RA,1) and the sixth resistance value (RA′,1), and (v.xxxv) applying a seventh electrical potential across the fifth and eighth contact elements for generating a seventh current in the surface portion at the first position, (v.xxxvi) measuring the seventh current through the fifth or the eighth contact element, (v.xxxvii) measuring a seventh voltage across the sixth and seventh contact elements, (v.xxxviii) calculating a seventh resistance value (RA,2) based on the seventh current and the seventh voltage, (v.xxxix) applying an eighth electrical potential across the sixth and seventh contact elements for generating an eighth current in the surface portion at the first position, (v.xl) measuring the eighth current through the sixth or the seventh contact element, (v.xli) measuring an eighth voltage across the fifth and eighth contact elements, (v.xlii) calculating an eighth resistance value (RA′,2) based on the eighth current and the eighth voltage, (v.xliii) calculating a fourth resistance mean (RAA′,2) of the seventh resistance value (RA,2) and the eighth resistance value (RA′,2), (v.xliv) defining a second relation including a fourth, fifth, and sixth parameter representing the first resistance mean (RBB′,1), the third resistance mean (RAA′,1), and a first pseudo sheet resistance (RP,1), respectively, (v.xlv) employing the first resistance mean (RBB′,1) and the third resistance mean (RAA′,1) as the fourth parameter and the fifth parameter, respectively, in the second relation for determining the sixth parameter representing the first pseudo sheet resistance (RP,1), (v.xlvi) defining a third relation including a seventh, eighth, and ninth parameter representing the second resistance mean (RBB′,2), the fourth resistance mean (RAA′,2), and a second pseudo sheet resistance (RP,2), respectively, (v.xlvii) employing the second resistance mean (RBB′,2) and the fourth resistance mean (RAA′,2) as the seventh parameter and the eighth parameter, respectively, in the third relation for determining the ninth parameter representing the second pseudo sheet resistance (RP,2), (v.xlviii) defining a fourth relation (gD) including a tenth, an eleventh, and a twelfth parameter representing the first pseudo sheet resistance (RP,1), the second pseudo sheet resistance (RP,2), and the distance between the first position and the electrical boundary, respectively, and (v.xlix) employing the first and the second pseudo sheet resistances (RP,1, RP,2) as the tenth and eleventh parameter, respectively, in the fourth relation (gD) for determining the twelfth parameter representing a value of the distance (y) between the first position and the electrical boundary.
The fourth relation may be equivalent to RP,1/RP,2=gD(y), where RP,1 represents the first pseudo sheet resistance, RP,2 represents the second pseudo sheet resistance, and gD represents a function including the distance y as a parameter, the function go defining a peak value at a specific distance, and the function gD increasing as a function of the distance below the specific distance and decreasing as a function of the distance above the specific distance, and the method may further comprise: (v.xlx) determining a distance by the method according to the first aspect of the present invention for representing an auxiliary distance, and (v.xlxi) comparing the auxiliary distance and the specific distance to determine if the distance is below or above the specific distance in the fourth relation.
The methods according to the first and second aspects of the present invention require only a single engagement between the probe and the test sample for accurately determining the distance.
An electrical boundary is to be understood as a boundary preventing a current from leaving the electrically conducting surface portion. Relation is here meant to encompass a single equation, a set of equations, a function, a set of functions, or any appropriate mathematical model of the setup used in the method that involves the specified parameters. The determining of the distance may encompass equation solving, regression analysis, comparisons with modelled or calibrated parameters, or any other mathematical technique appropriate for the specified relation.
The multi-point probe may have any number of contact elements equal to or greater than four, e.g. the multi-point probe may have twelve contact elements. However, four contact elements are employed independent on the total number of contact elements. A contact elements that is not used in the method may be positioned between two contact elements that are used in the method. The multi-point probe may be a probe described in any of WO2010115771A1, WO2010000265A1, WO2008110174A1, WO2007051471A1, WO005124371A1, or WO0003252A2. The contact elements may be in the form of cantilevers that extend from a probe body, e.g. as is shown in WO0003252A2. The test sample may be silicon wafer with a doped surface portion or a thin metal film defining the electrically conductive surface portion. The electrical boundary may be defined by a portion of the physical boundary of the electrically conductive surface portion, with a non-conducting region outside the boundary. Alternatively, the electrical boundary may be defined by a portion of the physical boundary of the test sample as a whole, provided the electrically conductive surface portion extends to the physical boundary.
The magnetic field may be generated by an electromagnet or a permanent magnet positioned on the opposite side of the test sample from the electrically conducting surface portion. The first, second, third, and fourth contact elements may be coupled to a multiplexer, which in turn is coupled to a current source for generating an electrical potential across two of the contact elements, an ammeter for measuring a current through the contact elements, and a voltmeter for measuring the voltage across two of the contact elements. A control unit may be coupled to the multiplexer to automatically control the multiplexer such that the currents can be generated and measured, and the voltages measured, as specified in the method.
The features described below may be employed for both the first and second aspects of the present invention.
The contact points may define a first line intersecting each of the contact points. The contact points of the first contact element, the second contact element, the third contact element, the fourth contact element, and the one or more additional contact elements may define a first line intersecting each of the contact points prior to the contact points contacting the test sample. The contact points of the first contact element, the second contact element, the third contact element, and the fourth contact element, may be located in the given order along the first line. The contact points of the fifth contact element, the sixth contact element, the seventh contact element, and the eighth contact element, may be located on and/or in the given order along the first line.
The spacing between the contact points of the first and second contact elements, the second and third contact elements, and the third and fourth contact elements may be approximately equal to a first spacing value (s1). The spacing between the contact points of the fifth and sixth contact elements, the sixth and seventh contact elements, and the seventh and eighth contact elements may be approximately equal to a second spacing value (s2).
In the alternative replacing steps (i.xii) to (i.xx) the first relation may be equivalent to ΔRBB′/ΔRBB′,2=fD(y,s1,s2), where ΔRBB′ represents the first resistance difference, ΔRBB′,2 represents the second resistance difference, and fD is a function including the distance y between the first position and the electrical boundary, the first spacing value s1, and the second spacing value s2.
The function fD(y,s1,s2) in said first relation ΔRBB′/ΔRBB′,2=fD(y,s1,s2) may be equivalent to (3*arctan(s1/2y)−arctan(3s1/2y))/(3*arctan(s2/2y)−arctan(3s2/2y)).
The second relation may be equivalent to exp(2π·RAA′,1/RP,1)−exp(2π·RBB′,1/RP,1)=1, where RP,1 is the first pseudo sheet resistance, RBB′,1 is the first resistance mean, and RAA′,1 is the third resistance mean; and the third relation may be equivalent to exp(2π·RAA′,2/RP,2)−exp(2π·RBB′,2/RP,2)=1, where RP,2 is the second pseudo sheet resistance, RBB′,2 is the second resistance mean, and RAA′,2 is the fourth resistance mean.
The contact points may define a first line intersecting each of the contact points, i.e. the contact points are on a common line.
In the alternative including step (iii.xlix) the fourth relation may be equivalent to RP,1/RP,2=gD(y,s1,s2), where RP,1 represents the first pseudo sheet resistance, RP,2 represents the second pseudo sheet resistance, and go represents a function including the distance y between the first position and the electrical boundary, the first spacing value s1, and the second spacing value s2.
In the alternative replacing step (iii.xlix) the first relation may be equivalent to ΔRBB′/ΔRBB′,2=fD(y,y2,s1,s2), where ΔRBB′ represents the first resistance difference, ΔRBB′,2 represents the second resistance difference, and fD is a function including the distance y and the additional distance y2 between the first position and the electrical boundary, the first spacing value s1, and the second spacing value s2.
The function fD(y,y2,s1,s2) in the first relation ΔRBB′/ΔRBB′,2=fD(y,y2,s1,s2) may be equivalent to (3*arctan(s1/2y)−arctan(3s1/2y))/(3*arctan(s2/2y2)−arctan(3s2/2y2)).
In the alternative replacing step (iii.xlix) the fourth relation may be equivalent to RP,1/RP,2=gD(y,y2,s1,s2), where RP,1 represents the first pseudo sheet resistance, RP,2 represents the second pseudo sheet resistance, and go represents a function including the distance y and the additional distance y2 between the first position and the electrical boundary, the first spacing value s1, and the second spacing value s2.
The electrical boundary may have an approximately linear portion and the distance between the first position and a point on the linear portion may be smaller than the distance between the first position and any point on the electrical boundary outside the linear portion. This means that the shortest distance from the first position to the electrical boundary is to a point on the linear portion.
The method may further comprise: (vi.i) orienting the multi-point probe to position the first line in a parallel relationship with the linear portion. The spacing between the contact points of the first and second contact elements, the second and third contact elements, and the third and fourth contact elements may be approximately equal.
In the alternative including steps (i.xii) to (i.xx) the first relation may be equivalent to ΔRCC′/ΔRBB′=f(y,s), where ΔRBB′ represents the first resistance difference, ΔRCC′ represents the second resistance difference, and f is a function including the distance y between the first position and the electrical boundary and the spacing s between the contact points as parameters.
The first resistance difference ΔRBB′ in the relation ΔRCC′/ΔRBB′=f(y,s) may be equivalent to ΔRBB′=2RH/π*(arctan(a/2y)+arctan(b/2y)+arctan(c/2y)−arctan((a+b+c)/2y)), where ΔRBB′ represents the additional first difference (ΔRBB′), y represents the distance (y), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements; and the second resistance difference ΔRCC′ in the relation ΔRCC′/ΔRBB′=f(y,s) may be equivalent to ΔRCC′=2RH/π*(arctan((a+b+c)/2y)+arctan(b/2y)−arctan((a+b)/2y)−arctan((b+c)/2y)) where ΔRCC′ represents the additional second difference (ΔRCC′), y represents the distance (y), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements.
In the alternative including steps (i.xii) to (i.xx) of the first aspect of the present invention the method may further comprise: (i.xxiii) applying a fifth electrical potential across the first and fourth contact elements for generating a fifth current in the surface portion at the first position, (i.xxiv) measuring the fifth current through the first or the fourth contact element, (i.xxv) measuring a fifth voltage across the second and third contact elements, (i.xxvi) calculating a fifth resistance value (RA) based on the fifth current and the fifth voltage, (i.xxvii) applying a sixth electrical potential across the second and third contact elements for generating a sixth current in the surface portion at the first position, (i.xxviii) measuring the sixth current through the second or the third contact element, (i.xxix) measuring a sixth voltage across the first and fourth contact elements, (i.xxx) calculating a sixth resistance value (RA′) based on the sixth current and the sixth voltage, (i.xxxi) calculating a third resistance difference (ΔRAA′) based on the difference between the fifth resistance value and the sixth resistance value, (i.xxxii) in defining the first relation (f) the first relation (f) further includes a fourth parameter representing the third resistance difference (ΔRAA′), and (i.xxxiii) in determining the distance (y) between the first position and the electrical boundary, the third resistance difference (ΔRAA′) is used as the fourth parameter in the first relation in addition to the first and the second resistance differences (ΔRBB′, ΔRCC′) may be used as the first and the second parameter, respectively.
This allows for the accurate measurements of the distance y with asymmetric probe that have non-equidistant contact points.
The contact points may be positioned in-line and the spacing between the contact points of the first and second contact elements and the spacing between the contact points of the second and third contact elements may be approximately equal. The contact points may be positioned in-line and the spacing between the contact points of the third and fourth contact elements may be different from the spacing between the contact points of the first and second contact elements. It has been found in simulations that this particular relative spacing of the contact probes gives accurate determinations of the distance to the electrical boundary.
The spacing between the contact points of the third and fourth contact element may be greater than the spacing between the contact points of the first and second contact element by a factor in one or more of the ranges 1.1-3.7, 1.2-3.3, 1.3-2.9, 1.4-2.5, 1.5-2.1, and 1.6-1.7, and/or in one of the ranges 1.2-1.3, 1.3-1.4, 1.4-1.5, 1.5-1.6, 1.6-1.7, 1.7-1.8, 1.8-1.9, 1.9-2.0, 2.0-2.2, 2.2-2.4, 2.4-2.6, 2.6-2.8, 2.8-3.0, 3.0-3.3, 3.3-3.6, 3.6-3.9, and/or approximately five divided by three, or by a factor in one or more of the ranges 1.2-3.8, 1.6-3.4, 1.8-3.2, 2.0-3.0, 2.2-2.8, and 2.4-2.6, and/or in one of the ranges 1.2-1.3, 1.3-1.4, 1.4-1.5, 1.5-1.6, 1.6-1.7, 1.7-1.8, 1.8-1.9, 1.9-2.0, 2.0-2.2, 2.2-2.4, 2.4-2.6, 2.6-2.8, 2.8-3.0, 3.0-3.3, 3.3-3.6, 3.6-3.9, and/or approximately five divided by two. It has been found in simulations that these particular relations give accurate determinations of the distance to the electrical boundary.
The spacing between the contact points of the first and second contact elements may be in one of the ranges 1-5 μm, 5-10 μm, 10-15 μm, 15-20 μm, 20-25 μm, 25-30 μm, 30-40 μm, 40-50 μm, 50-500 μm, and/or in one or more of the ranges 1-50 μm, 5-40 μm, 10-30 μm, 15-25 μm.
The first relation may be equivalent to (ΔRAA′+αΔRCC′)/ΔRBB′=f(y,a,b,c), where ΔRBB′ is the first resistance difference, ΔRCC′, is the second resistance difference, ΔRAA′ is the third resistance difference, α is a tuning factor in the range −10 to 10; and f is a function including the distance y between the first position and the electrical boundary and a the spacing between the contact points of the first and second contact elements, b the spacing between the contact points of the second and third contact elements, and c the spacing between the contact points of the third and fourth contact elements.
The tuning factor α may be approximately 1 or approximately −1.
The first resistance difference ΔRBB′ in the relation (ΔRAA′+αΔRCC′)/ΔRBB′=f(y,a,b,c) may be equivalent to ΔRBB′=2RH/π*(arctan(a/2y)+arctan(b/2y)+arctan(c/2y)−arctan((a+b+c)/2y)), where ΔRBB′ represents the additional first difference (ΔRAA′), y represents the distance (y), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements; the second resistance difference ΔRCC′ in the relation (ΔRAA′+αΔRCC′)/ΔRBB′=f(y,a,b,c) may be equivalent to ΔRCC′=2RH/π*(arctan((a+b+c)/2y)+arctan(b/2y)−arctan((a+b)/2y)−arctan((b+c)/2y)), where ΔRCC′ represents the additional second difference (ΔRCC′), y represents the distance (y), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements; and the third resistance difference ΔRAA′ in the relation (ΔRAA′+αΔRCC′)/ΔRBB′=f(y,a,b,c) may be equivalent to ΔRAA′=2RH/π*(arctan((a+b)/2y)−arctan(a/2y)−arctan((b+c)/2y)+arctan(c/2y)), where ΔRAA′ represents the additional third resistance difference (ΔRAA′), y represents the distance (y), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements. It has been found that this particular modelling gives accurate results over a wide range of measurement conditions.
The spacing between the contact points of the first and second contact elements may be in one or more of the ranges 0.1-100 μm, 1-90 μm, 10-80 μm, 20-70 μm, 30-60 μm, and 40-50 μm; and/or in one of the ranges 0.1-1 μm, 1-10 μm, 10-20 μm, 20-30 μm, 30-40 μm, 40-50 μm, 50-60 μm, 60-70 μm, 70-80 μm, 80-90 μm, 90-100 μm, or 100-500 μm.
The object of the invention is achieved according to a third aspect of the invention by a method for determining an electrical property at a first position on an electrically conducting surface portion of a test sample, the electrically conducting surface portion having an electrical boundary and the method comprising: (a) determining a distance (y) between the first position on the electrically conducting surface portion of the test sample and the electrical boundary of the electrically conducting surface portion according to any of the points 1 to 32, (b) defining a fifth relation including the electrical property and a fifteenth parameter representing the distance (y), and (c) employing the distance (y) as the fifteenth parameter in the fifth relation for determining the electrical property, or alternatively the method comprising: (a) determining an additional distance (y2) between the first position on the electrically conducting surface portion of the test sample and the electrical boundary of the electrically conducting surface portion according to any of the points 2 to 32, (b) defining a fifth relation including the electrical property and a fifteenth parameter representing the additional distance (y2), and (c) employing the additional distance (y2) as the fifteenth parameter in the fifth relation for determining the electrical property.
According to this method, only a single engagement is necessary between the probe and the test sample for accurately determining the electrical property.
An electrical boundary is to be understood as a boundary preventing a current from leaving the electrically conducting surface portion. Relation is here meant to encompass a single equation, a set of equations, a function, a set of functions, or any appropriate mathematical model of the setup used in the method that involves the specified parameters. The determining of the electrical property may encompass equation solving, regression analysis, comparisons with modelled or calibrated parameters, or any other mathematical technique appropriate for the specified relation.
The fifth relation may further including the spacing between the contact points of the first contact element, the second contact element, the third contact element, and/or the fourth contact element, and: (b′) in defining the fifth relation, the fifth relation may further include a sixteenth parameter representing the spacing between the contact points of the first contact element, the second contact element, the third contact element, and/or the fourth contact element, and (c′) in determining the electrical property, the spacing may be used as the sixteenth parameter in the fifth relation in addition to the distance (y) or the additional distance (y2).
The electrical property may be the Hall sheet resistance (RH) and the fifth relation (f1,f2) may further include a seventeenth parameter representing an additional first resistance difference (ΔRBB′), the method may further comprise: (d) applying an additional first electrical potential across the first and third contact elements for generating an additional first current in the surface portion at the first position, (e) measuring the additional first current through the first or the third contact element, (f) measuring an additional first voltage across the second and fourth contact elements, and (g) calculating an additional first resistance value (RB) based on the additional first current and the additional first voltage, or (g′) retaining the first resistance value (RB) from the determining of the distance as an additional first resistance value (RB), and (h) applying an additional second electrical potential across the second and fourth contact elements for generating an additional second current in the surface portion at the first position, (i) measuring the additional second current through the second or the fourth contact element, (j) measuring an additional second voltage across the first and third contact elements, and (k) calculating an additional second resistance value (RB′) based on the additional second current and the additional second voltage, or (k′) retaining the second resistance value (RB′) from the determining of the distance as an additional second resistance value (RB′), and (l) calculating the additional first resistance difference (ΔRBB′) based on the difference between the additional first resistance value and the additional second resistance value, or (l′) retaining the first resistance difference (ΔRBB′) from the determining of the distance as the additional first resistance difference (ΔRBB′).
The fifth relation may be equivalent to ΔRBB′=2RH/π*(arctan(a/2y)+arctan(b/2y)+arctan(c/2y)−arctan((a+b+c)/2y)) where ΔRBB′ represents the additional first difference (ΔRBB′), y represents the distance (y) or additional distance (y1), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements.
The electrical property may be the Hall sheet resistance (RH) and the fifth relation (f2,f3) may further include a seventeenth parameter representing an additional second resistance difference (ΔRCC′), the method may further comprise: (d) applying an additional third electrical potential across the first and second contact elements for generating an additional third current in the surface portion at the first position, (e) measuring the additional third current through the first or the second contact element, (f) measuring an additional third voltage across the third and fourth contact elements, and (g) calculating an additional third resistance value (RC) based on the additional third current and the additional third voltage, or in the alternative including steps (i.xii) to (i.xx): (g′) retaining the third resistance value (RC) from the determining of the distance as an additional third resistance value (RC), and (h) applying an additional fourth electrical potential across the third and fourth contact elements for generating an additional fourth current in the surface portion at the first position, (i) measuring the additional fourth current through the third or the fourth contact element, (j) measuring an additional fourth voltage across the first and second contact elements, and (k) calculating an additional fourth resistance value (RC′) based on the additional fourth current and the additional fourth voltage, or in the alternative including steps (i.xii) to (i.xx): (k′) retaining the fourth resistance value (RC′) from the determining of the distance as an additional fourth resistance value (RC′), and (l) calculating the additional second resistance difference (ΔRCC′) based on the difference between the additional third resistance value and the additional forth resistance value, or in the alternative including steps (i.xii) to (i.xx): (l′) retaining the second resistance difference (ΔRCC′) from the determining of the distance as the additional second resistance difference (ΔRCC′).
The fifth relation may be equivalent to ΔRCC′=2RH/π*(arctan((a+b+c)/2y)+arctan(b/2y)−arctan((a+b)/2y)−arctan((b+c)/2y)), where ΔRCC′ represents the additional second difference (ΔRCC′), y represents the distance (y) or additional distance (y1), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements.
The electrical property may be the Hall sheet resistance (RH) and the fifth relation (f1) may further include a seventeenth parameter representing an additional third resistance difference (ΔRAA′), the method may further comprise: (d) applying an additional fifth electrical potential across the first and fourth contact elements for generating an additional fifth current in the surface portion at the first position, (e) measuring the additional fifth current through the first or the fourth contact element, (f) measuring an additional fifth voltage across the second and third contact elements, and (g) calculating an additional fifth resistance value (RA) based on the additional fifth current and the additional fifth voltage, or (g′) retaining the fifth resistance value (RA) from the determining of the distance as an additional fifth resistance value (RA), and (h) applying an additional sixth electrical potential across the second and third contact elements for generating an additional sixth current in the surface portion at the first position, (i) measuring the additional sixth current through the second or the third contact element, (j) measuring an additional sixth voltage across the first and fourth contact elements, and (k) calculating an additional sixth resistance value (RA′) based on the additional sixth current and the additional sixth voltage, or (k′) retaining the sixth resistance value (RA′) from the determining of the distance as an additional fifth resistance value (RA′), and (l) calculating the additional third resistance difference (ΔRAA′) based on the difference between the additional fifth resistance value and the additional sixth resistance value, or (l′) retaining the third resistance difference (ΔRAA′) from the determining of the distance as the additional third resistance difference (ΔRAA′).
The fifth relation may be equivalent to ΔRAA′=2RH/π*(arctan((a+b)/2y)−arctan(a/2y)−arctan((b+c)/2y)+arctan(c/2y)), where ΔRAA′ represents the additional third resistance difference (ΔRAA′), y represents the distance (y) or additional distance (y1), a represents the spacing between the contact points of the first and second contact elements, b represents the spacing between the contact points of the second and third contact elements, and c represents the spacing between the contact points of the third and fourth contact elements.
The electrical property may be the sheet resistance (R0) and the fifth relation (g) may further include a seventeenth parameter representing a pseudo sheet resistance (RP), the method may further comprise: (d) applying an additional fifth electrical potential across the first and fourth contact elements for generating an additional fifth current in the surface portion at the first position, (e) measuring the additional fifth current through the first or the fourth contact element, (f) measuring an additional fifth voltage across the second and third contact elements, (g) calculating an additional fifth resistance value (RA) based on the additional fifth current and the additional fifth voltage, or (g′) retaining the fifth resistance value (RA) from the determining of the distance as an additional fifth resistance value (RA), and (h) applying an additional sixth electrical potential across the second and third contact elements for generating an additional sixth current in the surface portion at the first position, (i) measuring the additional sixth current through the second or the third contact element, (j) measuring an additional sixth voltage across the first and fourth contact elements, (k) calculating an additional sixth resistance value (RA′) based on the additional sixth current and the additional sixth voltage, or (k′) retaining the sixth resistance value (RA′) from the determining of the distance as an additional fifth resistance value (RA′), and (l) calculating a fifth resistance mean (RAA′) of the additional fifth resistance value (RA) and the additional sixth resistance value (RA′), and (d″) applying an additional first electrical potential across the first and third contact elements for generating an additional first current in the surface portion at the first position, (e″) measuring the additional first current through the first or the third contact element, (f″) measuring an additional first voltage across the second and fourth contact elements, (g″) calculating an additional first resistance value (RB) based on the additional first current and the additional first voltage, or (g″) retaining the first resistance value (RB) from the determining of the distance as an additional first resistance value (RB), and (h″) applying an additional second electrical potential across the second and fourth contact elements for generating an additional second current in the surface portion at the first position, (i″) measuring the additional second current through the second or the fourth contact element, (j″) measuring an additional second voltage across the first and third contact elements, (k″) calculating an additional second resistance value (RB′) based on the additional second current and the additional second voltage, or (k′″) retaining the second resistance value (RB′) from the determining of the distance as an additional second resistance value (R′B), and (l″) calculating a sixth resistance mean (RBB′) of the additional first resistance value (RB) and the additional second resistance value (RB′), and (m) defining a sixth relation including a eighteenth, nineteenth, and twentieth parameter representing the fifth resistance mean (RAA′), the sixth resistance mean (RBB′), and the pseudo sheet resistance (RP), respectively, (n) determining the pseudo sheet resistance (RP) by using the fifth resistance mean (RAA′) and the sixth resistance mean (RBB′) as the eighteenth parameter and the nineteenth parameter, respectively, in the sixth relation.
The sixth relation may be equivalent to exp(2π·RAA′/RP)−exp(2π·RBB′/RP)=1, where RP is the pseudo sheet resistance, RAA′ is the first resistance mean, and RBB′ is the second resistance mean.
The Hall sheet carrier density NHS may be calculated as: NHS=B/(ZeRH), where B is the magnetic field component perpendicular to the electrically conducting surface portion, Ze is the carrier charge (Z=±1), and RH the Hall sheet resistance. The mean Hall carrier mobility μH may be calculated as: μH=(ZRH)/(R0B), where Z=±1 depending on the type of carrier charge, RH the Hall sheet resistance, R0 the sheet resistance, and B is the magnetic field component perpendicular to the electrically conducting surface portion. If the magnetic field is generated by an electromagnet, the magnetic field component B may be calculated theoretically from the current through the electromagnet. If the magnetic field is generated by a permanent magnet, the magnetic field component B may be pre-determined by a suitable calibration.
The object of the invention is achieved according to a fourth aspect of the invention by an apparatus for determining a distance between a first position on an electrically conducting surface portion of a test sample and an electrical boundary of the electrically conducting surface portion, the apparatus comprising: a multi-point probe comprising a first contact element, a second contact element, a third contact element, a fourth contact element, contact element defining a contact point for establishing an electrical contact with the test sample, or a multi-point probe comprising a plurality of contact elements, and a control unit adapted for performing the method for determining a distance according to the first or second aspects of the present invention.
The object of the invention is achieved according to a fifth aspect of the invention by an apparatus for determining an electrical property at a first position on an electrically conducting surface portion of a test sample comprising: a multi-point probe comprising a first contact element, a second contact element, a third contact element, and a fourth contact element, each contact element defining a contact point for establishing an electrical contact with the test sample, or a multi-point probe comprising a plurality of contact elements, and a control unit adapted for performing the method for determining an electrical property at a first position on an electrically conducting surface portion of a test sample according to the third aspect of the present invention.